If $(x + k)$ is a factor of the polynomial $x^{2} - 2 x - 15$ and $x^{2} + a$ . Find $k$ and $a$ .

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Solution

$x (k + 2) (x + k)) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ x^{2} - 2 x - 15 x^{2} + k x______(- k - 2) x - 15 - (k + 2) x - k (k + 2)_k (k + 2) - 15$

$(x + k)$ is a factor of $x^{2} - 2 x - 15$ and $x^{2} + a$
$\Rightarrow k (k + 2) - 15 = 0$
$\Rightarrow k^{2} + 2 k - 15 = 0$
$\Rightarrow k^{2} + 5 k - 3 k - 15 = 0$
$\Rightarrow k (k + 5) - 3 (k + 5) = 0$
$\Rightarrow k = 3 o r - 5$
if $k = 3$

$x - 3 (x + 3)) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ x^{2} + a x^{2} + 3 x_- 3 x + a - 3 x - 9__9 + a$
$9 + a = 0$
$a = - 9$
if $k = - 5$

$x + 5 (x - 5)) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ x^{2} + a x^{2} - 5 x_5 x + a 5 x - 25__________25 + a$
$25 + a = 0$
$\Rightarrow a = - 25$
$∴$ The solution are
$(k, a) = (3, - 9)$ and $(- 5, - 25)$

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